*(English)*Zbl 0721.30035

This paper is devoted to generalizing the trace-squared (conjugation) invariant of Möbius transformations with complex entries to elements of a certain set of Clifford matrices ${\mathcal{C}}^{n}$. These were first considered by *K. Th. Vahlen* [Math. Ann. 55, 585-593 (1902)] and later by *L. V. Ahlfors* [see e.g. Ann. Acad. Sci. Fenn., Ser. A I 105, 15-27 (1985; Zbl 0586.30045)]. The Clifford matrices considered here are slightly different from those obtained by Ahlfors and provide a generalization to orientation-reversing maps. The main theorem is that, for each $k=0,1,2,\xb7\xb7\xb7,n+2$, the ${T}_{k}$ function (the definition is too complicated to give here)

is a conjugation invariant. Here ${\mathcal{C}}^{n}$ is the algebra of Clifford numbers generated by n roots of -1. The conjugation is by Clifford matrices. Among the properties of the invariant ${T}_{k}$ are the following:

(i) ${T}_{k}(-g)=T\left(g\right)$ for all $g\in {\mathcal{C}}^{n}$. Thus ${T}_{k}$ is well-defined on the projectivization of ${\mathcal{C}}^{n}\xb7$

(ii) ${T}_{k}\left(g\right)=0$ for all k odd (resp. even) if g is orientation preserving (resp. reversing)

(iii) $g\in {\mathcal{C}}^{n}$ is hyperbolic if and only if ${T}_{k}\left(g\right)<0$ for the largest k so that ${T}_{k}\left(g\right)\ne 0\xb7$

(iv) $g\in {\mathcal{C}}^{n}$ is elliptic if and only if the quadratic form $N\left(z\right):=z\overline{z}$, when restricted to the kernel of Id-r(g) is not positive semidefinite. Here r takes g from the isometries of the ball model of hyperbolic space to the hyperboloid model.

##### MSC:

30G35 | Functions of hypercomplex variables and generalized variables |